 In this video, I'm going to do a couple of examples of how to translate linear functions, just a couple of examples. I'm going to do one example where I translate a function to the left, and I'm going to do another example where I think the example is where I translate it down. I'll have to go to the next slide to figure that out. So again, here are the directions. Let g of x be the indicated transformation of f of x. Write the rule for g of x. Now, when the directions say write the rule for g of x, it's basically saying write the equation for it. Write the function itself. So what we're going to do, let g be the indicated transformation. So what I'm going to do is I'm going to take f of x, and I'm going to transform it. I'm going to transform it, and then the new one is going to be g of x. Sometimes these directions can be a little confusing. So let's take this function, and we're going to horizontally translate it left three units. So now what I'm going to do is I'm not going to draw f of x. I'm actually going to wait and draw g of x once I get there. I'm going to draw my new function. So what I'm going to do is I'm going to show you the notation behind this, and then we're going to draw the new shape. So the horizontal translation, three units to the left. What's it going to look like? I'm going to take my original function. I'm going to change it. This is what this notation means. Take my original function, change it by. Now, since this is a horizontal translation, I'm going to add or subtract directly to the x. That's why you got a big parentheses here. So now that if I'm going three units to the left, it's actually going to be opposite of what you think. What I commonly call opposite of what you think. If you're doing a horizontal translation, if you do anything inside parentheses like this, it's always going to be opposite of what you think. Now, if you think of a number line, if you go left three units, you think you're going back three units. You're going down. You're subtracting three. Well, actually, if you ever do something inside parentheses like this, it's actually going to be opposite of what you think. So I'm actually going to add three directly to the x. I'm going to add three directly to the x. That is going to give me a shift left of three units. So what I'm going to do is I'm going to take my new function. G of x, this is my new function, is going to be the old one. But I'm going to take the x and replace it with x plus 3. So now you can see that here it's the same notation here. So my new function is going to be the old function, except for instead of an x, it's going to be an x plus 3. So I'm going to take my old function, which is x minus 2. But instead of the x, I'm going to put an x plus 3. Instead of the x, I'm going to put an x plus 3. I still got that minus 2 there at the back. So see, notice here that I used parentheses. I took out this x and put in the x plus 3. OK, so that's what I mean by take out the x and put in the x plus 3. So my new G of x function, my new function, is going to be. Now, I don't need these parentheses there. There's no reason to have them. There's no number out front to distribute or anything like that. So actually what this is going to be is x plus 1. Positive 3 and a negative 2, it makes a positive 1. So that's the equation for my new line. All right, so what I'm going to do is I'm going to take this new line. I'm going to graph it. So I got 1 right there for a y-intercept. And my slope is a 1 right there. We don't write it, but there is a 1 right there. So I got a slope of positive 1. Positive what? Positive 1, positive 1, positive 1, positive 1, positive 1, positive 1. There we go. I don't need that many points, but it's always good to put too many than not enough. OK, so that's my new function, G. That's the new function, G. OK, now that's what the graph of it looks like. So this is my answer. Now what I'm also going to do, just to really bring some full understanding to this, is now I'm actually going to graph my old function. I'm going to graph f of x. Now what I'm going to do is I'm going to change my color. And this f of x is actually going to be blue. OK, so what I'm going to do is I'm going to graph this, x minus 2. So minus 2 for my y-intercept right there, and then 1 for my slope. OK, 1 for my slope right here. OK, a couple of points here. Connect the dots. And there we go. OK, that's my f function. Now notice here that, OK, so what I wanted to do is I wanted to move. I wanted to take my f of x function and horizontally translate it three units to the left. So did I do that? So here's my f of x function. Did I move all my points three units to the left? Well, here's 1 point. 1, 2, 3. Yes, I did move it three units to the left. Well, let's take another point right here. 1, 2, 3. Yes, I did move it three units to the left. So we can see that, even though the function is to look a lot different now, OK, we've got different numbers here, moving it three units to the left, that did actually happen, OK? So we did do that correctly, OK? And here's the notation of how to find the new equation. Or you can just graph it to figure out what the new equation is going to be, OK? So there's a couple of different ways you can do that. Now, the only thing that we had to do here was write the rule for g of x. So I could have stopped right here with just with the function, just with this new equation here. I didn't have to graph it. But I wanted to graph it to show you guys kind of exactly what we're doing here. We have a little bit better understanding. OK, next example. Let g of x, it's basically the same thing, let g of x be the indicated transformation of f of x. Write the rule for g of x. So again, we're writing a new rule. OK, now we're going to take f of x minus 2, so it's the same function, and we're going to vertically translate it down three units. Vertically translate it down three units. OK, so what we're going to do is we're going to take my function, still using the blue pen here, take my function, change it by just simply taking the function, and if I want to go down three units, I'm just going to subtract three to it. That's it. Now notice I am not subtracting inside these parentheses, so I'm going to flip back real quick. When I did my translations left, I added three directly inside these parentheses. But now, on my next example, now since I'm doing a vertical translation, since I'm moving down three units totally different, I'm not adding or subtracting inside the parentheses. I'm doing it outside now. I'm doing it outside now. All right, so now what I'm going to do is I'm going to write the new equation. So g of x is equal to my old function, f of x, but instead of just having f of x, I want to move it down three. So now I'm going to subtract three from it. So notice I'm using the same notation here. So what I'm going to do is I'm going to take this function and just subtract three from it. So take x minus two and then just simply subtract three from it. So take my function, x minus two, and then simply just subtract three from it. There it is. And so my new function, g of x, is going to be equal to x minus five. x minus five. OK, so now what I'm going to do is I'm going to very quickly, I'm going to graph the new one, g, and then the old one, f. I'm going to graph those quick. And I'm actually going to switch up the colors here. The blue is going to be g this time, and then I'll flip my colors for black for f. OK, so x minus five, so one, two, three, four, five right there at the very bottom of this graph I have here. And then a slope of one. Slope of one. Slope of one. Slope of one. Get lots of different points here. Straight line, try to get that as straight as you can. And there's my g function. There's my new g function. OK, let's compare that to the old one, change my colors here, and change that to the old one. x minus two was the equation. All right, so I have minus two for my y intercept, and a slope of one, slope of one, slope of one, OK, slope of one. Put a bunch of different points in here. Make sure we get a nice straight line. Nice straight line there. And that's my f function. Now the question is, did I move this down three units? OK, so take your f function, did I go down three? One, two, three. Yep, I went down three. Let's pick another point. Here's another one. One, two, three. Yes, I did go down three units. OK, so I did do that correctly. They do that correctly. So this right here, this g of x, this is my new rule. That's my new function for my new equation, my new line here. OK, that's just a couple of examples of translating different functions. The first one was translating left, and then this one is translating down.